Networked Synchronization Control of Coupled ... - Semantic Scholar
1468
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 40, NO. 6, DECEMBER 2010
Networked Synchronization Control of Coupled Dynamic Networks With Time-Varying Delay Yingchun Wang, Member, IEEE, Huaguang Zhang, Senior Member, IEEE, Xingyuan Wang, and Dongsheng Yang
Abstract—This paper is concerned with the networked synchronization control problem of coupled dynamic networks (CDNs) with time-varying delay. First, both the data packet dropouts and network-induced delays are taken into account in the synchronization controller design. A Markovian jump process is induced to describe the packet dropouts. The network-induced delays are interval time varying and depend on the Markovian jump modes. A new closed-loop coupled dynamic error system (CDES) with Markovian jump parameters and interval time-varying delays is constructed. Second, using the Kronecker product technique and the stochastic Lyapunov method, a delay-dependent sufficient criterion of stochastic stability is obtained for the closed-loop CDES, which also guarantees that the CDNs are stochastically synchronized. Finally, a simulation example is given to demonstrate the effectiveness of the proposed result. Index Terms—Coupled dynamic networks (CDNs), Kronecker product, Markovian jump, networked synchronization control, time-varying delay.
I. I NTRODUCTION
S
INCE THE pioneering work of Watts and Strogatz [1], the dynamics analysis of coupled dynamic networks [(CDNs) including coupled neural networks] has been paid much attention due to its important application in various fields such as physics, technology, and the life sciences [2]–[7]. In [3], a sufficient condition of global asymptotical synchronization for the coupled neural networks was derived by using the Hermitian matrix theory and Lyapunov functional method. In [4], some sufficient conditions were proposed for the global exponential synchronization of the coupled neural networks with constant delays by linear matrix inequality (LMI) techniques. The global exponential synchronization problem for an array of neural networks with coupled mixed delays (discrete and distributed Manuscript received May 21, 2009; revised October 11, 2009 and November 13, 2009; accepted December 24, 2009. Date of publication March 1, 2010; date of current version November 17, 2010. This work was supported in part by the National Natural Science Foundation of China under Grants 60904017, 60774048, 60904046, and 5097708, by the Program for Cheung Kong Scholars, National Basic Research Program of China, under Grant 2009CB320601, by the Startup Fund of Doctor of Liaoning, China, under Grant 20081020, and by the China postdoctoral science foundation, under Grant 20070421063. This paper was recommended by Associate Editor R. Selmic. Y. Wang, H. Zhang, and D. Yang are with the School of Information Science and Engineering and the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China (e-mail: [email protected]; [email protected]; [email protected]). X. Wang is with the School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, China (e-mail: wangxy@ dlut.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2010.2040273
delays) was studied in [5]. Moreover, those synchronization research works were also extended to the stochastic CDNs. The stochastic coupling and disturbances were considered for the general delayed discrete-time networks in [6]. The CDNs with Markovian jump parameters and mixed delays were studied in [7], and some exponential synchronization results were proposed. Aside from the dynamic analysis of CDNs, the synchronization control for CDNs is also an important issue to be studied due to its potential applications in secure communication, biology and information science, etc. So far, the adaptive synchronization control is a main control approach for the CDNs (see [8]–[12]), but some unfavorable effects of data transmission between the controller and CDNs have not been considered. Therefore, further researches are indispensable for the synchronization control of CDNs. On the other hand, much effort has been devoted for the networked control systems due to its many advantages (such as simple installation, low cost and maintenance, and high reliability) (see [13] and [14]). Some important networked modeling and control methods have been proposed for the linear or nonlinear systems (see [15]–[24]). In [15], an H∞ networked control method for linear uncertain systems was proposed by employing a new Lyapunov–Krasovskii functional. In [16], a multiple successive delay component-based networked control modeling approach was proposed and an H∞ controller was designed. In [17], a robust controller was designed under the assumption that the data packet dropouts and the network-induced delays satisfy a known probability distribution. A sampleddata control approach was presented for the linear networked control systems in [18]. The packet-based control approaches for discrete- and continuous-time systems were proposed in [19] and [20], respectively. In [21] and [22], a stochastic Markovian jump controller was designed under the assumption that the data packet dropouts on both sides of the controller satisfy two different Markovian processes. Considering the nonlinear networked control systems, some fuzzy-based control approaches were proposed (see [23] and [24]). It is emphasized that, due to CDNs’ spatial distribution characteristic in practical application, the data transmission between the CDNs and controllers be implemented under the network environment. Therefore, the networked synchronization control problem of CDNs should be considered. However, this kind of results on networked synchronization control for master–slave systems are not found in the existing literature, not to mention the CDNs with time-varying delay, which motivates our research. In this paper, we deal with the networked synchronization control problem for an array of CDNs with time-varying delay. Based on the sampled-data approach, a new closed-loop
1083-4419/$26.00 © 2010 IEEE
WANG et al.: NETWORKED SYNCHRONIZATION CONTROL OF CDNs WITH TIME-VARYING DELAY
1469
CDES with Markovian jump parameters and interval timevarying delays is constructed under the assumptions that the data packet dropouts satisfy a known Markovian process and that the network-induced delays (from sensors to controllers and from controllers to actuators) are interval time varying for each Markovian jump mode. Then, by applying the stochastic Lyapunov stability theory and the Kronecker product technique, a delay-dependent sufficient condition is obtained such that the closed-loop CDES is stochastically stable, which also guarantees the stochastic synchronization of the CDNs. Moreover, the sufficient condition is in terms of the LMIs, which can be resolved effectively by the LMI toolbox in Matlab. A simulation example is given to show the effectiveness of the result in this paper. II. P ROBLEM F ORMULATION AND C ONTROL A NALYSIS A. Problem Formulation In this paper, the CDNs with time-varying delay [25] are formulated as follows: Fig. 1. Schematic diagram of networked synchronization control systems.
y˙ l (t) = − Ayl (t) + Bg (yl (t)) + Cgd (yl (t − d(t))) N + G1lm Γ1 ym (t) +
m=1 N
G2lm Γ2 ym (t − d(t))+Ul (t)
Next, we consider the networked synchronization control problem of CDNs with time-varying delay (see Fig. 1). The isolated dynamic network is (1)
m=1
where l = 1, 2, . . . , N , yl (t) = [yl1 (t), yl2 (t), . . . , yln (t)]T ∈ Rn is the state vector of the ith node, A is a constant matrix, B and C are the connection and delayed-connection weight matrices, respectively, Γ1 and Γ2 are the constant inner coupling diagonal matrices of the nodes, and Gs = [Gslm ]N ×N , s = 1, 2, is the out-coupling matrix of the network defined as follows: If there is a connection from node m to node l (l = m), then Gslm > 0 and Gsll = − m=l Gslm ; otherwise, Gslm = 0. d(t) is a time-varying delay in the state which satisfies 0 < dm ¯ ≤ d(t) ≤ dM
˙ ≤μ 0, lim →0 o( )/ = 0. Here, γij ≥ 0 is the tran sition rate from i to j if i = j, while γii = − N i=j,j=1 γij . Then, the synthesis controller with network-induced delay and data packet dropouts is U(t) = (IN ⊗ Ki )e (t − τi (t)) ,
for r(t) = i
(7)
and the closed-loop CDES is e(t) ˙ = − (IN ⊗ A)e(t) + (IN ⊗ B)f (e(t)) + (IN ⊗ C)fd (e (t − d(t))) + (G1 ⊗ Γ1 )e(t) + (G2 ⊗ Γ2 )e (t − d(t)) + (IN ⊗ Ki )e (t − τi (t)) (8)
(6)
where τ (t) is piecewise linear with derivative τ˙ (t) = 1 for t = lk h + τk and τ (t) is discontinuous at the points t = lk h + τk . It is clear that h ≤ τk ≤ τ (t) < lk+1 h − lk h + τk+1 . Consider the case of the data packet dropouts in the transmission process (see the right side of Fig. 2). It shows that r data packets drop out between the sampling instants lk+1 h and lk+2 h. Whenever packet dropout happens from the sampler to the controller or from the controller to the actuator, the last control signal in the holder will hold until the next signal is received. Δ Let S = {l1 , l2 , . . .}, a subsequence of {1, 2, . . .}, denote the sequence of time points of successful data transmissions from Δ the sampler to the actuator and N = maxlk ∈S (lk+1 − lk ) be the maximum packet-loss upper bound. Then, at time t, t ∈ [lk h + τk , lk+1 h + τk+1 ), k = 1, 2, . . ., the packet-loss process r(t) can be defined as r(t) = lk+1 − lk ,
Δ
which takes value in the finite state space S¯ = {1, 2, . . . , N }. We assume that the packet-loss process r(t) satisfies a Markovian process taking values in a finite state space S¯ = {1, 2, . . . , N } with generator Γ = (γij )N ×N given by γij + o( ), if i = j P {r(t + ) = j|r(t) = i} = 1 + γij + o( ), if i = j
where τi (t) is the time delay depending on the Markovian jump mode i. Assume that there exist some constants τim ¯ and τiM such that h ≤ τim ≤ τ (t) ≤ τ , i = 1, . . . , N . Let ϕT (t) = ¯ i iM T T T [ϕ1 (t), ϕ2 (t), . . . , ϕN (t)] be a vector-valued continuous function of initial system states on the interval [−dM , 0] that does not depend on the Markovian process r(t). For simplicity, we define this initial function ϕ(t) with ϕ. Remark 1: Since the sampler and the controller in this paper are clock driven and synchronized, the size of the minimum induced delay is larger than one sampling period, i.e., h ≤ min(τi (t), i = 1, . . . , N ). It is different from the common case of the clock-driven sampler but eventdriven controller, which is difficult to realize in practical applications. A new closed-loop CDES is constructed with Markovian jump parameters and mode-dependent interval time-varying delays, which is essentially a kind of new networked control models and can be analyzed by the stochastic theory. Note that Markovian jump-delayed systems have been researched in the past decade and some
WANG et al.: NETWORKED SYNCHRONIZATION CONTROL OF CDNs WITH TIME-VARYING DELAY
important results were achieved (see [26]–[29]). The linear delay interval systems and the nonlinear systems with timevarying delays were considered in [28] and [29], respectively. However, the nonlinear Markovian jump systems with interval time-varying delay has not been well considered. Our aim is to propose a new Lyapunov–Krasovskii functional-based stability analysis approach for the closed-loop CDES with Markovian jump parameters and interval time-varying delays (8). Next, let us give some definitions and useful lemmas. Definition 1: The closed-loop CDES (8) is said to be stochastically stable if, for continuous function ϕ defined on [−dM , 0], there exists a positive constant ρ(ϕ) satisfying ρ(0) = 0, such that ⎧ t ⎫ ⎨ ⎬ eT (s, ϕ, r(s)) e (s, ϕ, r(s)) ds | ϕ ≤ ρ(ϕ) lim E t→∞ ⎩ ⎭
1471
3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD). T ), Ξ = (ulm )N ×N , Lemma 3: Let z T = (z1T , z2T , . . . , zN n×n T T T T , and y = (y1 , y2 , . . . , yN ) with zk , yk ∈ Rn , k = P ∈R 1, 2, . . . , N . If Ξ = ΞT and each row sum of Ξ is zero, then z T (ΞT Ξ) ⊗ P y =− ukl ukm (zl − zm )T P (yl − ym ). 1≤l<m≤N 1≤k≤N T Proof: Let Q = Ξ Ξ. We know that Q = (qlm )N ×N = ( 1≤k≤N ukl ukm )N ×N is also symmetric and each row sum of Q is zero. From [7, Lemma 1] ulm (zl − zm )T P (yl − ym ) z T (Ξ ⊗ P )y = − 1≤l<m≤N
0
where E is the mathematical expectation operator. Definition 2: If the closed-loop CDES (8) is stochastically stable, then it is said that the CDNs (1) are stochastically synchronized, i.e., lim E |el (t, r(t), ϕl )|2 2 = lim E yl t; t0 , Yl0 − x(t; t0 , x0 ) = 0
t→∞
t→∞
t−β 1
υ˙ T (s)J υ(s) ˙ ds
t−β2
≤−
= z T (Q ⊗ P )y ukl ukm (zl − zm )T P (yl − ym ). =− 1≤l<m≤N 1≤k≤N
This completes the proof.
where yl (t; t0 , Yl0 ) (1 ≤ l ≤ N ) is the solution of (1); Yl0 and x0 are the initial states of (1) and (3), respectively. Note that, when the CDNs (1) or (2) achieve synchronization, the coupled terms and the control input should vanish, i.e., (G1 ⊗ Γ1 )y(t) = 0, (G2 ⊗ Γ2 )y(t − d(t)) = 0, and U(t) = 0. Lemma 1 (cf. [30]): For any symmetric constant matrix J ∈ Rn×n , J ≥ 0, scalars β1 and β2 with β1 < β2 , and a vectorvalued function υ(t) : [−β2 , −β1 ] → Rn , t ∈ R+ , such that the integration in the following is well defined −(β2 − β1 )
we have z T (ΞT Ξ) ⊗ P y
υ(t − β1 ) υ(t − β2 )
T
J −J
−J J
υ(t − β1 ) . υ(t − β2 )
Lemma 2 (cf. [31]): The Kronecker product has the following properties: 1) (αA) ⊗ B = A ⊗ (αB); 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C;
⎡
Φil,m
Φi1,1,l,m ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ =⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
Φi1,2,l,m Φi2,2,l,m ∗ ∗ ∗ ∗ ∗
Φi1,3,l,m Φi2,3,l,m Φi3,3,l,m ∗ ∗ ∗ ∗
III. M AIN R ESULT In this section, we are to give the main result of this paper. ⎡ ⎤ N −1 −1 ··· −1 N − 1 ··· −1 ⎥ ⎢ −1 ¯1 = Denote U = ⎣ ⎦, K ··· ··· ··· ··· −1 −1 ··· N − 1 ¯ 2 = (K1 + K2 )/2, W ¯ 1 = (W T W2 + (K1T K2 + K2T K1 )/2, K 1 T ¯ W2 = (W1 + W2 )/2, τm W2 W1 )/2, ¯ = min(τim ¯ ), τM = max(τiM ), and γ = max(|γii |), i = 1, . . . , N . Theorem 1: The closed-loop CDES (8) is stochastically stable, which ensures that the CDNs are stochastically synchronized under the controller (7), if there exist matrices Pi > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, R3 > 0, invertible ¯ i , and scalars ε1 > 0 and ε2 > 0 such that the followmatrix K ing LMIs hold for 1 ≤ l < m ≤ N and i = 1, . . . , N , with (9) shown at the bottom of the page:
Φi1,4,l,m Φi2,4,l,m Φi3,4,l,m Φi4,4,l,m ∗ ∗ ∗
Φi1,5,l,m Φi2,5,l,m Φi3,5,l,m Φi4,5,l,m Φi5,5,l,m ∗ ∗
Q2 > γQ1
(10)
R3 > γ(τM − τm ¯ )R2
(11)
0 0 0 Φi4,6 0 Φi6,6 ∗
⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 40, NO. 6, DECEMBER 2010
Networked Synchronization Control of Coupled Dynamic Networks With Time-Varying Delay Yingchun Wang, Member, IEEE, Huaguang Zhang, Senior Member, IEEE, Xingyuan Wang, and Dongsheng Yang
Abstract—This paper is concerned with the networked synchronization control problem of coupled dynamic networks (CDNs) with time-varying delay. First, both the data packet dropouts and network-induced delays are taken into account in the synchronization controller design. A Markovian jump process is induced to describe the packet dropouts. The network-induced delays are interval time varying and depend on the Markovian jump modes. A new closed-loop coupled dynamic error system (CDES) with Markovian jump parameters and interval time-varying delays is constructed. Second, using the Kronecker product technique and the stochastic Lyapunov method, a delay-dependent sufficient criterion of stochastic stability is obtained for the closed-loop CDES, which also guarantees that the CDNs are stochastically synchronized. Finally, a simulation example is given to demonstrate the effectiveness of the proposed result. Index Terms—Coupled dynamic networks (CDNs), Kronecker product, Markovian jump, networked synchronization control, time-varying delay.
I. I NTRODUCTION
S
INCE THE pioneering work of Watts and Strogatz [1], the dynamics analysis of coupled dynamic networks [(CDNs) including coupled neural networks] has been paid much attention due to its important application in various fields such as physics, technology, and the life sciences [2]–[7]. In [3], a sufficient condition of global asymptotical synchronization for the coupled neural networks was derived by using the Hermitian matrix theory and Lyapunov functional method. In [4], some sufficient conditions were proposed for the global exponential synchronization of the coupled neural networks with constant delays by linear matrix inequality (LMI) techniques. The global exponential synchronization problem for an array of neural networks with coupled mixed delays (discrete and distributed Manuscript received May 21, 2009; revised October 11, 2009 and November 13, 2009; accepted December 24, 2009. Date of publication March 1, 2010; date of current version November 17, 2010. This work was supported in part by the National Natural Science Foundation of China under Grants 60904017, 60774048, 60904046, and 5097708, by the Program for Cheung Kong Scholars, National Basic Research Program of China, under Grant 2009CB320601, by the Startup Fund of Doctor of Liaoning, China, under Grant 20081020, and by the China postdoctoral science foundation, under Grant 20070421063. This paper was recommended by Associate Editor R. Selmic. Y. Wang, H. Zhang, and D. Yang are with the School of Information Science and Engineering and the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China (e-mail: [email protected]; [email protected]; [email protected]). X. Wang is with the School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, China (e-mail: wangxy@ dlut.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2010.2040273
delays) was studied in [5]. Moreover, those synchronization research works were also extended to the stochastic CDNs. The stochastic coupling and disturbances were considered for the general delayed discrete-time networks in [6]. The CDNs with Markovian jump parameters and mixed delays were studied in [7], and some exponential synchronization results were proposed. Aside from the dynamic analysis of CDNs, the synchronization control for CDNs is also an important issue to be studied due to its potential applications in secure communication, biology and information science, etc. So far, the adaptive synchronization control is a main control approach for the CDNs (see [8]–[12]), but some unfavorable effects of data transmission between the controller and CDNs have not been considered. Therefore, further researches are indispensable for the synchronization control of CDNs. On the other hand, much effort has been devoted for the networked control systems due to its many advantages (such as simple installation, low cost and maintenance, and high reliability) (see [13] and [14]). Some important networked modeling and control methods have been proposed for the linear or nonlinear systems (see [15]–[24]). In [15], an H∞ networked control method for linear uncertain systems was proposed by employing a new Lyapunov–Krasovskii functional. In [16], a multiple successive delay component-based networked control modeling approach was proposed and an H∞ controller was designed. In [17], a robust controller was designed under the assumption that the data packet dropouts and the network-induced delays satisfy a known probability distribution. A sampleddata control approach was presented for the linear networked control systems in [18]. The packet-based control approaches for discrete- and continuous-time systems were proposed in [19] and [20], respectively. In [21] and [22], a stochastic Markovian jump controller was designed under the assumption that the data packet dropouts on both sides of the controller satisfy two different Markovian processes. Considering the nonlinear networked control systems, some fuzzy-based control approaches were proposed (see [23] and [24]). It is emphasized that, due to CDNs’ spatial distribution characteristic in practical application, the data transmission between the CDNs and controllers be implemented under the network environment. Therefore, the networked synchronization control problem of CDNs should be considered. However, this kind of results on networked synchronization control for master–slave systems are not found in the existing literature, not to mention the CDNs with time-varying delay, which motivates our research. In this paper, we deal with the networked synchronization control problem for an array of CDNs with time-varying delay. Based on the sampled-data approach, a new closed-loop
1083-4419/$26.00 © 2010 IEEE
WANG et al.: NETWORKED SYNCHRONIZATION CONTROL OF CDNs WITH TIME-VARYING DELAY
1469
CDES with Markovian jump parameters and interval timevarying delays is constructed under the assumptions that the data packet dropouts satisfy a known Markovian process and that the network-induced delays (from sensors to controllers and from controllers to actuators) are interval time varying for each Markovian jump mode. Then, by applying the stochastic Lyapunov stability theory and the Kronecker product technique, a delay-dependent sufficient condition is obtained such that the closed-loop CDES is stochastically stable, which also guarantees the stochastic synchronization of the CDNs. Moreover, the sufficient condition is in terms of the LMIs, which can be resolved effectively by the LMI toolbox in Matlab. A simulation example is given to show the effectiveness of the result in this paper. II. P ROBLEM F ORMULATION AND C ONTROL A NALYSIS A. Problem Formulation In this paper, the CDNs with time-varying delay [25] are formulated as follows: Fig. 1. Schematic diagram of networked synchronization control systems.
y˙ l (t) = − Ayl (t) + Bg (yl (t)) + Cgd (yl (t − d(t))) N + G1lm Γ1 ym (t) +
m=1 N
G2lm Γ2 ym (t − d(t))+Ul (t)
Next, we consider the networked synchronization control problem of CDNs with time-varying delay (see Fig. 1). The isolated dynamic network is (1)
m=1
where l = 1, 2, . . . , N , yl (t) = [yl1 (t), yl2 (t), . . . , yln (t)]T ∈ Rn is the state vector of the ith node, A is a constant matrix, B and C are the connection and delayed-connection weight matrices, respectively, Γ1 and Γ2 are the constant inner coupling diagonal matrices of the nodes, and Gs = [Gslm ]N ×N , s = 1, 2, is the out-coupling matrix of the network defined as follows: If there is a connection from node m to node l (l = m), then Gslm > 0 and Gsll = − m=l Gslm ; otherwise, Gslm = 0. d(t) is a time-varying delay in the state which satisfies 0 < dm ¯ ≤ d(t) ≤ dM
˙ ≤μ 0, lim →0 o( )/ = 0. Here, γij ≥ 0 is the tran sition rate from i to j if i = j, while γii = − N i=j,j=1 γij . Then, the synthesis controller with network-induced delay and data packet dropouts is U(t) = (IN ⊗ Ki )e (t − τi (t)) ,
for r(t) = i
(7)
and the closed-loop CDES is e(t) ˙ = − (IN ⊗ A)e(t) + (IN ⊗ B)f (e(t)) + (IN ⊗ C)fd (e (t − d(t))) + (G1 ⊗ Γ1 )e(t) + (G2 ⊗ Γ2 )e (t − d(t)) + (IN ⊗ Ki )e (t − τi (t)) (8)
(6)
where τ (t) is piecewise linear with derivative τ˙ (t) = 1 for t = lk h + τk and τ (t) is discontinuous at the points t = lk h + τk . It is clear that h ≤ τk ≤ τ (t) < lk+1 h − lk h + τk+1 . Consider the case of the data packet dropouts in the transmission process (see the right side of Fig. 2). It shows that r data packets drop out between the sampling instants lk+1 h and lk+2 h. Whenever packet dropout happens from the sampler to the controller or from the controller to the actuator, the last control signal in the holder will hold until the next signal is received. Δ Let S = {l1 , l2 , . . .}, a subsequence of {1, 2, . . .}, denote the sequence of time points of successful data transmissions from Δ the sampler to the actuator and N = maxlk ∈S (lk+1 − lk ) be the maximum packet-loss upper bound. Then, at time t, t ∈ [lk h + τk , lk+1 h + τk+1 ), k = 1, 2, . . ., the packet-loss process r(t) can be defined as r(t) = lk+1 − lk ,
Δ
which takes value in the finite state space S¯ = {1, 2, . . . , N }. We assume that the packet-loss process r(t) satisfies a Markovian process taking values in a finite state space S¯ = {1, 2, . . . , N } with generator Γ = (γij )N ×N given by γij + o( ), if i = j P {r(t + ) = j|r(t) = i} = 1 + γij + o( ), if i = j
where τi (t) is the time delay depending on the Markovian jump mode i. Assume that there exist some constants τim ¯ and τiM such that h ≤ τim ≤ τ (t) ≤ τ , i = 1, . . . , N . Let ϕT (t) = ¯ i iM T T T [ϕ1 (t), ϕ2 (t), . . . , ϕN (t)] be a vector-valued continuous function of initial system states on the interval [−dM , 0] that does not depend on the Markovian process r(t). For simplicity, we define this initial function ϕ(t) with ϕ. Remark 1: Since the sampler and the controller in this paper are clock driven and synchronized, the size of the minimum induced delay is larger than one sampling period, i.e., h ≤ min(τi (t), i = 1, . . . , N ). It is different from the common case of the clock-driven sampler but eventdriven controller, which is difficult to realize in practical applications. A new closed-loop CDES is constructed with Markovian jump parameters and mode-dependent interval time-varying delays, which is essentially a kind of new networked control models and can be analyzed by the stochastic theory. Note that Markovian jump-delayed systems have been researched in the past decade and some
WANG et al.: NETWORKED SYNCHRONIZATION CONTROL OF CDNs WITH TIME-VARYING DELAY
important results were achieved (see [26]–[29]). The linear delay interval systems and the nonlinear systems with timevarying delays were considered in [28] and [29], respectively. However, the nonlinear Markovian jump systems with interval time-varying delay has not been well considered. Our aim is to propose a new Lyapunov–Krasovskii functional-based stability analysis approach for the closed-loop CDES with Markovian jump parameters and interval time-varying delays (8). Next, let us give some definitions and useful lemmas. Definition 1: The closed-loop CDES (8) is said to be stochastically stable if, for continuous function ϕ defined on [−dM , 0], there exists a positive constant ρ(ϕ) satisfying ρ(0) = 0, such that ⎧ t ⎫ ⎨ ⎬ eT (s, ϕ, r(s)) e (s, ϕ, r(s)) ds | ϕ ≤ ρ(ϕ) lim E t→∞ ⎩ ⎭
1471
3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD). T ), Ξ = (ulm )N ×N , Lemma 3: Let z T = (z1T , z2T , . . . , zN n×n T T T T , and y = (y1 , y2 , . . . , yN ) with zk , yk ∈ Rn , k = P ∈R 1, 2, . . . , N . If Ξ = ΞT and each row sum of Ξ is zero, then z T (ΞT Ξ) ⊗ P y =− ukl ukm (zl − zm )T P (yl − ym ). 1≤l<m≤N 1≤k≤N T Proof: Let Q = Ξ Ξ. We know that Q = (qlm )N ×N = ( 1≤k≤N ukl ukm )N ×N is also symmetric and each row sum of Q is zero. From [7, Lemma 1] ulm (zl − zm )T P (yl − ym ) z T (Ξ ⊗ P )y = − 1≤l<m≤N
0
where E is the mathematical expectation operator. Definition 2: If the closed-loop CDES (8) is stochastically stable, then it is said that the CDNs (1) are stochastically synchronized, i.e., lim E |el (t, r(t), ϕl )|2 2 = lim E yl t; t0 , Yl0 − x(t; t0 , x0 ) = 0
t→∞
t→∞
t−β 1
υ˙ T (s)J υ(s) ˙ ds
t−β2
≤−
= z T (Q ⊗ P )y ukl ukm (zl − zm )T P (yl − ym ). =− 1≤l<m≤N 1≤k≤N
This completes the proof.
where yl (t; t0 , Yl0 ) (1 ≤ l ≤ N ) is the solution of (1); Yl0 and x0 are the initial states of (1) and (3), respectively. Note that, when the CDNs (1) or (2) achieve synchronization, the coupled terms and the control input should vanish, i.e., (G1 ⊗ Γ1 )y(t) = 0, (G2 ⊗ Γ2 )y(t − d(t)) = 0, and U(t) = 0. Lemma 1 (cf. [30]): For any symmetric constant matrix J ∈ Rn×n , J ≥ 0, scalars β1 and β2 with β1 < β2 , and a vectorvalued function υ(t) : [−β2 , −β1 ] → Rn , t ∈ R+ , such that the integration in the following is well defined −(β2 − β1 )
we have z T (ΞT Ξ) ⊗ P y
υ(t − β1 ) υ(t − β2 )
T
J −J
−J J
υ(t − β1 ) . υ(t − β2 )
Lemma 2 (cf. [31]): The Kronecker product has the following properties: 1) (αA) ⊗ B = A ⊗ (αB); 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C;
⎡
Φil,m
Φi1,1,l,m ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ =⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
Φi1,2,l,m Φi2,2,l,m ∗ ∗ ∗ ∗ ∗
Φi1,3,l,m Φi2,3,l,m Φi3,3,l,m ∗ ∗ ∗ ∗
III. M AIN R ESULT In this section, we are to give the main result of this paper. ⎡ ⎤ N −1 −1 ··· −1 N − 1 ··· −1 ⎥ ⎢ −1 ¯1 = Denote U = ⎣ ⎦, K ··· ··· ··· ··· −1 −1 ··· N − 1 ¯ 2 = (K1 + K2 )/2, W ¯ 1 = (W T W2 + (K1T K2 + K2T K1 )/2, K 1 T ¯ W2 = (W1 + W2 )/2, τm W2 W1 )/2, ¯ = min(τim ¯ ), τM = max(τiM ), and γ = max(|γii |), i = 1, . . . , N . Theorem 1: The closed-loop CDES (8) is stochastically stable, which ensures that the CDNs are stochastically synchronized under the controller (7), if there exist matrices Pi > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, R3 > 0, invertible ¯ i , and scalars ε1 > 0 and ε2 > 0 such that the followmatrix K ing LMIs hold for 1 ≤ l < m ≤ N and i = 1, . . . , N , with (9) shown at the bottom of the page:
Φi1,4,l,m Φi2,4,l,m Φi3,4,l,m Φi4,4,l,m ∗ ∗ ∗
Φi1,5,l,m Φi2,5,l,m Φi3,5,l,m Φi4,5,l,m Φi5,5,l,m ∗ ∗
Q2 > γQ1
(10)
R3 > γ(τM − τm ¯ )R2
(11)
0 0 0 Φi4,6 0 Φi6,6 ∗
⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥
